full spectra
DESCRIPTION
Full spectra presentation by Prof TumaTRANSCRIPT
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FULL SPECTRA AS A TOOL FOR ANALYSIS OF A SHAFT ROTATING IN
JOURNAL BEARINGS
Ji Tma & Jan Bilo17. listopadu 15, Ostrava Poruba
Czech [email protected] & [email protected]
VB TU Ostrava Faculty of Mechanical Engineering
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Outline
Bently Nevada Rotorkit Instrumentation arrangement Orbit plot, one-side versus two-side spectrum RPM profile & displacement time history RMS full multispectrum Fluid induced instability Bently and Muszynska model Equation of motion Shaft/fluid wedge bearing/system as a servomechanism Vibration modes
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Bently Nevada Rotorkit
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Bently Nevada Rotorkit - detail
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Instrumentation arrangementProximity probes
y(t) imaginary partx(t) real part
Journal bearing
Journal
Shaft
Fluid lubrication
Displacement x(t)
Displacementy(t)
Complex coordinate of the shaft centre position:
X
Y
-
Orbit plot
+ X
Y
-
A
B = A* (Re)
(Im)
Both the vectors A and B are rotating in opposite direction at the same frequency .
+ X
Y
-
A
B (Re)
(Im)
Real harmonic function of time
Complex harmonic function of time
Ellipse
-
One-side versus two-side spectrum
Orbit plot
-1,5
0,0
1,5
-1,5 0,0 1,5
X (Re)
Y (Im
)
Time history
-1,5
0,0
1,5
0,0 0,5 1,0
Time [s]
X, Y
(Re,
Im)
Autospectrum
0,00,20,40,60,81,01,2
-5 -4 -3 -2 -1 0 1 2 3 4 5
Frequency [Hz]R
MS
X +
j Y
Time history
-1,5-1,0-0,50,00,51,01,5
0,0 0,5 1,0Time [s]
X (R
e)
Autospectrum
0,0
0,2
0,4
0,6
0,8
-5 -4 -3 -2 -1 0 1 2 3 4 5Frequency [Hz]
RM
S X
Autospectrum
0,0
0,2
0,4
0,6
0,8
0 1 2 3 4 5Frequency [Hz]
RM
S X
Two-side spectrum One-side spectrum
Two-side spectrum
symmetry
non-symmetry
Fourier Transform
FT
FT
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RPM profile & displacement time historyTachometer
1000
1500
2000
2500
0 5 10 15Time [s]
RPM
Time History
-200-100
0100200300400500
0 5 10 15Time [s]
mic
ron X
Y;;
Steady-state vibration
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RMS full multispectrum of signal x(t) + j y(t)
-100 -9
0 -80 -70 -60 -50 -40 -30 -20 -10 0 10 2
0 30 40 50 60 70 80 90 100
16931935
21832378
24062212
19751727
010203040506070
RMS m
Frequency [Hz]
RPM
Autospectrum : X + jY
60-7050-6040-5030-4020-3010-200-10
0.475 ord 1.0 ord 2.0 ord
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Fluid induced instability
Time : X (Resampled) ; Y (Resampled)
-150-100-50
050
100150
0 1 2 3 4 5 6 7 8 9 10Revolution [-]
m
X (Resampled)Y (Resampled)
Orbit plot
-150-100
-500
50100150
-150 0 150
X [m]
Y [
m]
Autospectrum : X (Resampled) + jY (Resampled)
0,475 ord
0
20
40
60
80
-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0
Order [-]
RM
S [
m]
Self- excited whirl vibration
Low rotational speed vibration
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Full order-multispectrum of signalx(t) + j y(t)
-2,0
-1,8
-1,6
-1,4
-1,2
-1,0
-0,8
-0,6
-0,4
-0,2 0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
1,8
2,0
174619072057219423252384240123922299218120451896173415551350
RMS dB / ref 1 m
Order [-]
RPMAutospectrum : X (Resampled) + Y (Resampled)
35-4030-3525-3020-2515-2010-155-100-5
Proportionality of the whirl vibrationfrequency to the shaft rotational speed
0.475 ord 1.0 ord
-0.475 ord
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Bently and Muszynska model
avgv BearingShaft
= avg
v
Average fluid angular velocity
Fluid circumferential velocity ratio
rotrotrot DK rrF &+=
Fluid forces acting on the rotor
Spring and damper systemrotating at the angular frequency
Fluid wedge
-
Fluid forces in stationary coordinate
Fluid forcesTransform to stationary coordinates
r Rotating
Stationary
t
( )( ) ( )tjj
tj
rotrot
rot
==
expexp
rrrrr&&
rrrF += jDDK &
Direct Quadrature
Tangentionalforce
Direct force
Gravity force
An example of force balance
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Equation of motionAngular velocity is completely independent of the rotor angular velocity
Equation of motion
Perturbation balance force( )( )+= tjmruonPerturbati exp2F
( ) ( )( )+=++ tjmrjDKDM u exp2rrr &&&
( ) ( )( ) ( )+
=jDMK
jmrj u22 expexpA
Amplitude A and phase of the rotor shaft centre-line rotating at the angular velocity
( )( )+= tjexpArSolution takes a form
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Shaft/fluid wedge bearing/system as a servomechanism
Direct stiffness
Quadrature stiffness
( )2+=
=
MjDKjKDirect
( ) DjjKQuadrature =
( ) ( )20 +
=MKjD
DjG
Open-loop frequency transfer function
( )jKDirect1
( )jKQuadrature
Rotorload
Fluid wedgesupport
+
-
Rotor shaft eccentricity position
Positive feedback
Equation of motion
-
Closed-loop stability margin
=> Bently and Muszynskathreshold
( ) 10 =CritjG
According to the Nyquist stability criterion, a margin of stability is resulting from
=> Mechanical resonanceMKCrit =
2
CritCrit =
=
MKCrit
=> Fluid resonance
( )jG0
Real
(-1,0)
stable
margin
unstable0=
0=
0=
Crit=
Crit
Imag
Nyquist plot ofin complex plane
Crit>
Crit
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Vibration modes
-120-80-40
04080
120
0 5 10 15 Time [s]
mic
ron
X;;
Crit=
Crit
Unbalance effect Unbalance effect
0,1
1
10
100
0 5 10 15 20
Time [s]
Mag
nitu
de
x2 x1 x2 x3 x
( ) 10 =jG
0
10
20
30
40
50
60
70
1000 1300 1600 1900 2200 2500
RPM
Mag
nitu
de
xcoast down
run up
( ) 2=MtyeccentriciK=>
Magnitude self-control
Crit=
Harmonic envelopes
Fluid Induced vibrationEccentricity
Wal
l
Cen
tre
Stiffness K
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Whirl & WhipFluid Induced Instabilities
Whirl vibration
Whip vibration
Bently & Muszynskathreshold
Measurement range for this paper
An example Subharmonic
Harmonics of rotational frequency
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Conclusion
This presentation describes using full spectra for rotor system diagnostics
The full spectrum is a good tool for studying rotor instability in journal bearings
The presentation demonstrates whirl vibration and the independence of the ratio relating the precession speed to the shaft rotational speed with respect to the shaft absolute rotational speed
Bently and Muszynska model gives explanation of the rotor instability